Contextual Considerations in Probabilistic Situations: An Aid or a Hindrance?
We examine the responses of secondary school teachers to a probability task with an infinite sample space. Specifically, the participants were asked to comment on a potential disagreement between two students when evaluating the probability of picking a p
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Abstract We examine the responses of secondary school teachers to a probability task with an infinite sample space. Specifically, the participants were asked to comment on a potential disagreement between two students when evaluating the probability of picking a particular real number from a given interval of real numbers. Their responses were analyzed via the theoretical lens of reducing abstraction. The results show a strong dependence on a contextualized interpretation of the task, even when formal mathematical knowledge is evidenced in the responses.
Consider the conversation between two students presented in Fig. 1 and a teacher’s potential responses. The scenario is a familiar one—two students grappling with opposing responses to a probability task. The task itself is less familiar—the likelihood of choosing a particular event from an infinite sample space. In this chapter, we consider the specific mathematics embedded in the task in Fig. 1, and analyze the responses of practicing and prospective secondary school mathematics teachers as they addressed the scenario. Unlike conventional probability tasks, such as tossing a coin or throwing a die, a special feature of the presented task is that the embedded experiment—picking “any real number”—cannot be carried out. We begin by examining different aspects of probability tasks, the contexts in which they are presented, and the associated interpretations.
1 On Platonic vs. Contextualized Chernoff (2011) distinguished between platonic and contextualized sequences in probability tasks related to relative likelihood of occurrences. He suggested that platonic sequences are characterized by their idealism. For example, when considering A. Mamolo (B) University of Ontario Institute of Technology, 11 Simcoe St. N., Oshawa, ON, Canada, L1H 7L7 e-mail: [email protected] R. Zazkis Simon Fraser University, 8888 University Drive, Burnaby, BC, Canada, V5A 1S6 E.J. Chernoff, B. Sriraman (eds.), Probabilistic Thinking, Advances in Mathematics Education, DOI 10.1007/978-94-007-7155-0_34, © Springer Science+Business Media Dordrecht 2014
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The following conversation occurred between Damon and Ava, two Grade 12 students. Imagine you are their teacher and that they have asked for your opinion. They approach you with the following: Damon: I asked Ava to pick any real number between 1 and 10, write it down, and keep it a secret. Then we wanted to figure out what the probability was that I would guess right which number she picked. Ava: Right. And we did this a few times. The first time I picked 5, and Damon guessed it right on the first try. The next time, since he said “any real number,” I picked 4.7835. He never got that one. Damon: So, we tried to figure out the probabilities. I think that the probability of picking 5 is larger than the probability of picking 4.7835. Ava thinks the probability is the same for both numbers. Who’s right? Please consider and respond to the following questions: 1. What is the probability that Damon would guess corre
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